The question of extending these ideas to more general Riemannian manifolds was first investigated by Sogge in the case of the sphere [1]} and for general compact manifolds [2]}. This ultimately led to Theorem 5.1.1 in his textbook [3]}, which is optimal for general manifolds of finite geometry. A difficult and interesting problem is to understand the relation between the geometry of the manifold and the boundedness properties of the spectral projectors.
